Finish by midnight on Sunday if your whole class is doing parallelograms.

IMPORTANT – it does not really matter what score you get, because the main thing is that you think hard about the problems... and then examine the solution sheet to learn from your mistakes.

1. Kaprekar constant

This is a video from the hugely successful maths video channel Numberphile. The channel has received over 300 million views, so I would encourage you to explore some of its other videos if you want to learn more about maths.

Professor Roger Bowley from the University of Nottingham explains why 6,174 is such a special number. Watch it and answer the questions below.

In short:

Pick any 4 digit number (as longs the digits are not ALL the same).

Put the digits in ascending order and descending order to create two numbers.

Subtract the small number from the big number, to create a next stage number.

Then repeat the process with the new number

Eventually you end up with 6,174.

1 mark

1.1. Start with the number 4,321. Reverse the digits, take the small number from the big number and what is the result? Clue: it is between 3,000 and 4,000.

Correct Solution: 3,087

4,321 – 1,234 = 3,087

1 mark

1.2. Repeat the process, starting with your previous answer – remember to rearrange the digits into ascending order and descending order before subtracting the smaller number from the larger number. What is the result? Clue: it is between 8,000 and 9,000.

Correct Solution: 8,352

8,730 – 0378 = 8,352.

2 marks

1.3. Repeat the process, starting with your previous answer – remember to rearrange the digits into ascending order and descending order before subtracting the smaller number from the larger number. What is the result?

Correct Solution: 6,174

8,532 – 2,358 = 6,174 = Kaprekar’s constant.

3 marks

1.4. Every 4-digit number ends up as 6,174 if we follow the instructions above, but some numbers take longer to get there than others. How many stages does it take for 5,200?

Clue: To get you started, after the first stage we reach 5,200 – 0025 = 5,175.

2

3

7

9

10

5,200 – 0,025 = 5,175

7,551 – 1,557 = 5,994

9,954 – 4,599 = 5,355

5,553 – 3,555 = 1,998

9,981 – 1,899 = 8,082

8,820 – 0,288 = 8,532

8,532 – 2,358 = 6,174

3 marks

1.5. 6,174 is Kaprekar’s constant for 4 digits, but if we apply the same process for 3-digit numbers, then they all end up by transforming into a different number. Start with the number 972 and try to find out what constant it becomes after a few steps.

Show Hint (–1 mark)

–1 mark

The answer is between 400 and 500.

Correct Solution: 495

972 – 279 = 693

963 – 369 = 594

954 – 459 = 495

2. Wizard of Oz

If you have watched “The Wizard of Oz”, then you might not have spotted the maths that appears towards the end of the film. The Scarecrow famously wants a brain, but the best that the Wizard of Oz can do is to give him a diploma. This does not give the Scarecrow any knowledge or intelligence, but it does give him confidence.

Listen to the Scarecrow’s version of Pythagoras’s Theorem.

The Scarecrow recites an incorrect version of the Pythagorean Theorem: "The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.”

1 mark

2.1 Instead of “square roots”, what should the Scarecrow have said?

Squares

Cubes

Lengths

Widths

1 mark

2.2 Instead of “isosceles”, what should the Scarecrow have said?

Scalene

Right-angle

Equilateral

Symmetrical

3. Junior Maths Challenge Problem (UKMT)

If you are a Year 8 student, then it is likely that you will be taking part in the United Kingdom Maths Trust (UKMT) competition known as the Junior Maths Challenge (JMC). If you do particularly well, you might earn yourself a gold, silver or bronze certificate, but you will have to work hard as you will be competing against students from across the country.

Your teachers will help you prepare for this national maths competition, but in each week's Parallelogram we will always include one UKMT Junior Maths Challenge question.

3 marks

3.1 Gill went for a five-hour walk. Her average speed was between 3 km/h and 4 km/h.

As her average speed is between 3 km/h and 4 km/h, she walks between 15 km and 20km. Of the options given, only 19 km is in this range.

I hope you enjoyed the first Parallelogram of the year. There will be more next week, and the week after, and the week after that. So check your email or return to the website on Thursday at 3pm.

In the meantime, you can find out your score, the answers and go through the answer sheet as soon as you hit the SUBMIT button below.

When you see your % score, this will also be your reward score. As you collect more and more points, you will collect more and more badges. Find out more by visiting the Rewards Page after you hit the SUBMIT button.

It is really important that you go through the solution sheet. Seriously important. What you got right is much less important than what you got wrong, because where you went wrong provides you with an opportunity to learn something new.